a) Write the total number of permutations P(n, n) of a set of n different objects taken 'n' at a time ? [1] b) In the expansion of (1 + x)^n , what is the sum of the binomial coefficients ? [1] c)…

a) The total number of permutations of $n$ distinct objects taken $n$ at a time is given by: $$P(n,n)=n!$$ This is because we arrange all $n$ objects in a sequence, and the number of ways to do this is the factorial of $n$.

b) In the expansion of $(1+x{)}^n$, the sum of the binomial coefficients is obtained by setting $x=1$: $$(1+1{)}^n=2^n$$ Thus, the sum of the coefficients is $2^n$.

c) The general term in the expansion of $(1+x{)}^n$ is given by the binomial theorem: $$T_{k+1}=\left(\genfrac{}{}{0ex}{}{n}{k}\right)x^k$$ where $k=0,1,2,\dots ,n$.

d) The series expansion for ${\log }_e(1+x)$ for $|x|<1$ is derived from its Maclaurin series: $${\log }_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$$

e) The series representing $e^{-1}$ is obtained by substituting $x=-1$ in the exponential series: $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$$ Thus, $$e^{-1}=1-1+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots$$